Wee welcome all those who are interested to join us on Tuesdays, 3:30–4:30 PM, Ryerson Laboratory, Room 358, unless noted otherwise.

Everyone is also invited to attend the Learning Seminar on Geometric Analysis, organized by Prof. Andre Neves. More information on http://www.math.uchicago.edu/~geometric_analysis/learning

Lucas Ambrozio, IAS

** Comparing the total volume of a closed Riemannian manifold and the geometric invariants related to its minimal hypersurfaces
:**
The variational methods used to find closed embedded minimal hypersurfaces in a closed Riemannian manifold allow one to define several notions of "systole" and "width", which can be regarded as functionals on the space of Riemannian metrics. In this talk, we will be interested in the properties of some of these functionals on the space of unit volume metrics, focusing on the case of three dimensional spheres and real projective spaces. In particular, we will look for (sharp) upper bounds on certain subsets of metrics and describe necessary conditions satisfied by local maxima. This is joint work with Rafael Montezuma.

Nicolau Aiex, UBC

** TBA
:**
TBA

Davi Maximo, UPenn

** On the topology and index of minimal surfaces
:**
For an immersed minimal surface in R^3, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resolves several conjectures made by J. Choe and D. Hoffman concerning the classification of low-index minimal surfaces: we show that there are no complete two-sided immersed minimal surfaces in R^3 of index two, complete embedded minimal surface with index three, or complete one-sided minimal immersion with index one. This is joint work with Otis Chodosh.

Alessandro Carlotto, IAS/ETH-Zurich

** Constrained deformations of positive scalar curvature metrics
:**
I will present a series of results concerning the interplay between two different curvature conditions, in the special case when these are given by pointwise inequalities on the scalar curvature of a manifold, and the mean curvature of its boundary. Such results lie at two conceptual levels: on the one hand at the level of compatibility (i.e. is it possible to simultaneously satisfy the bounds, and what are the resulting topological implications), on the other hand at the level of moduli space structure (i.e. what can one say about the homotopy type of the associated space of metrics, when not empty, quotiented by the diffeomorphism group of the background manifold).
In particular, we give a complete topological characterization of those compact 3-manifolds that support Riemannian metrics of positive scalar curvature and mean-convex boundary and, in any such case, we prove that the associated moduli space of metrics is path-connected. Furthermore, we show how our methods can be refined so to construct continuous paths of positive scalar curvature metrics with minimal boundary, and to obtain analogous conclusions in that context as well.
Our work relies on a combination of earlier fundamental contributions by Gromov-Lawson and Schoen-Yau, on the smoothing procedure designed by Miao, and on the interplay of Perelman's Ricci flow with surgery and conformal deformation techniques introduced by Codá Marques in dealing with the closed case. This lecture is based on joint work with Chao Li (Princeton University).

Harrison Pugh, Notre Dame

** TBA
:**
TBA

Ben Sharp, University of Leeds

** Global estimates for harmonic maps from surfaces
:**
A celebrated theorem of F. Hélein guarantees that a weakly harmonic map from a two-dimensional domain is always smooth. The proof is of a local nature and assumes that the Dirichlet energy is sufficiently small; under this condition it is possible to re-write the harmonic map equation using a suitably chosen frame which uncovers non-linearities with more favourable regularity properties (so-called div-curl or Wente structures). We will prove a global estimate for harmonic maps without assuming a small energy bound, utilising a powerful theory introduced by T. Rivière. Along the way we will highlight the relevance of Wente-type estimates in neighbouring areas of geometric analysis, and hint as to why the analogous higher-dimensional global estimate remains a challenging open problem. This is a joint work with Tobias Lamm.

Henrik Matthiesen, The University of Chicago

** The systole of large genus minimal surfaces in positive Ricci curvature :**
We prove that the systole (or more generally, any $k$-th homology systole) of a minimal surface in an ambient three manifold of positive Ricci curvature tends to zero as the genus of the minimal surfaces becomes unbounded. This is joint work with Anna Siffert.

Xin Zhou, UCSB

** Multiplicity One Conjecture in Min-max theory :**
I will present a recent proof of the Multiplicity One Conjecture in Min-max theory. This conjecture was raised by Marques and Neves. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are all two-sided and have multiplicity one. As direct corollaries, it implies the generalized Yau's conjecture for such manifolds with positive Ricci curvature, which says that there exist infinitely many pairwise non-isometric minimal hypersurfaces, and the Weighted Morse Index Bound Conjecture by Marques and Neves.

Tristan Riviere, ETH

** The Cost of the Sphere Eversion and the 16\pi Conjecture :**
How much does it cost...to knot a closed simple curve ? To cover the sphere twice ? to realize such or such homotopy class ? ...etc.
All these questions consisting of assigning a "canonical" number and possibly an optimal "shape" to a given topological operation are known to be mathematically very rich and to bring together notions and techniques from topology, geometry and analysis.
In this talk we will concentrate on the operation consisting of turning inside out the 2 sphere in the 3 dimensional space. Since Smale's proof in 1959 of the existence of such an operation the search for effective realizations of such eversions has triggered a lot of fascination and works in the math community. The absence in nature of matter that can interpenetrate and the quasi impossibility, up to the advent of virtual imaging, to experience this deformation is maybe the reason for the difficulty to develop an intuitive approach on the problem.
We will present the optimization of Sophie Germain conformally invariant elastic energy for the eversion.
Our efforts will finally bring us to consider more closely an integer number together with a mysterious minimal surface.

Vanderson Lima, UFRGS

** Stable Minimal Surfaces and the topology of 3-Manifolds :**
Meeks, Perez and Ros conjectured that a closed Riemannian 3-manifold which does not contain any closed embedded stable minimal surface must be diffeomorphic to a quotient of the 3-sphere. In this talk we show a counter example for this conjecture. We also discuss how to correct the conjecture so that it holds true.

Casey Kelleher, Princeton University

** INDEX-ENERGY ESTIMATES FOR YANG-MILLS CONNECTIONS AND EINSTEIN METRICS :**
We prove a conformally invariant estimate for the index of Schrodinger operators acting on vector bundles over four-manifolds, related to the classical Cwikel-Lieb-Rozenblum estimate. Applied to Yang-Mills connections we obtain a bound for the index in terms of its energy which is conformally invariant, and captures the sharp growth rate. Furthermore we derive an index estimate for Einstein metrics in terms of the topology and the Einstein-Hilbert energy. Lastly we derive conformally invariant estimates for the Betti numbers of an oriented four-manifold with positive scalar curvature.

Giuseppe Tinaglia, King's College

** The geometry of constant mean curvature surfaces in Euclidean space :**
In this talk I will begin by reviewing classical geometric properties of constant mean curvature surfaces, H>0, in R^3. I will then talk about several more recent results for surfaces embedded in R^3 with constant mean curvature, such as curvature and radius estimates for simply-connected surfaces embedded in R^3 with constant mean curvature. Finally I will show applications of such estimates including a characterisation of the round sphere as the only simply-connected surface embedded in R^3 with constant mean curvature and area estimates for compact surfaces embedded in a flat torus with constant mean curvature and finite genus. This is joint work with Meeks.

Gregory Chambers, Rice University

** Geometric stability of the Coulomb energy :**
Suppose that $A$ is a measurable subset of $\mathbb{R}^3$ of finite measure. The Coulomb energy of $A$ is the double integral over $A$ of $1/|x-y|$, and is maximized when $A$ is a ball. If the Coulomb energy is close to maximal, then is $A$ geometrically close to a ball? We will answer this question, and will compare it to the quantitative isoperimetric inequality. We will also discuss the analogous situation in higher dimensions. This is joint work with Almut Burchard

Yaiza Canzani, UNC

** Understanding the growth of Laplace eigenfunctions :**
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Using the description of concentration, we obtain quantitative improvements on the known bounds in a wide variety of settings.

Yu Wang, Northwestern University

** Sharp estimate of global Coulomb gauge in dimension 4 :**
Consider a principle SU(2)-bundle P over a compact 4-manifold M and a W^{1,2}-connection A of P satisfying \|F_A\|_{L^2(M)}\le \Lambda. Our main result is the existence of a global section \sigma: M\to P with controllably many singularities such that the connection form \sigma^*A satisfies the Coulomb equation d^*(\sigma^*A)=0 and moreover admits a sharp estimate.
In this talk, we first recall some preliminaries and then outlines the proof. If time allows we shall elaborate on the ideas to overcome the main difficulties in this problem, which include an epsilon-regularity theorem for the Coulomb gauge, an annular-bubble regions decomposition for the curvature, and studying the singularity behavior of the Coulomb gauge on each annular and bubble region.

** ASYMPTOTIC BEHAVIOUR OF SOLUTIONS FOR A COUPLED ELLIPTIC SYSTEM IN THE PUNCTURED BALL :**
Our main goal is to study the asymptotic behavior near an isolated singularity of local solutions for certain strongly coupled critical elliptic systems that, from the viewpoint of conformal geometry, are pure extensions of Yamabe-type equations.
There has been considerable interest in recent years in proving compactness results for this type of systems since such type of problems provides a natural background for the interplay between geometry and asymptotic analysis.
We prove a sharp result on the removability of the isolated singularity for all components of the solutions when the dimension is less than or equal to five and minus the potential of the operator in the system is cooperative.

Lucas Ambrosio, University of Warwick

**
Sequences of minimal surfaces with bounded index in three-manifolds
:**
We explore a few consequences of the bubbling analysis developed by Buzano and Sharp, giving a detailed description of how a sequence of closed embedded minimal surfaces of bounded index in a three-manifold degenerates as we pass to a "converging" subsequence. In particular, a few new compactness result are obtained. This is a joint work with Reto Buzano (QMUL), A. Carlotto (ETH) and B. Sharp (Leeds).

Costante Bellettini, University College of London

**
Stable constant-mean-curvature hypersurfaces: regularity and compactness.
:**
This talk describes a recent joint work of the speaker with N. Wickramasekera (Cambridge). The work develops a regularity theory, with an associated compactness theorem, for weakly defined hypersurfaces (codimension 1 integral varifolds) of a smooth Riemannian manifold that are stationary and stable on their regular parts for volume preserving ambient deformations. The main regularity theorem gives two structural conditions on such a hypersurface that imply that, away from a set of codimension 7 or higher, the hypersurface is locally either a single smoothly embedded disk or precisely two smoothly embedded disks intersecting tangentially. Easy examples show that neither structural hypothesis can be relaxed. An important special case is when the varifold corresponds to the boundary of a Caccioppoli set, in which case the structural conditions can be considerably weakened. An "effective version" of the compactness theorem has been (a posteriori) established in collaboration with O. Chodosh and N. Wickramasekera.

Daniel Agress, University of California Irvine

**
Existence results for the nonlinear Hodge minimal surface energy.
:**
The nonlinear Hodge minimal surface energy, first studied by Sibner and Sibner in the 1970's, has applications to minimal surfaces, bounded variation functions, and the Born Infeld theory of electromagnetism. In this talk, we will prove an existence and nonexistence result for minimizers of the energy. In particular, we show that for a compact Riemannian manifold and cohomology class $[\alpha] \in H^k(M)$, minimizers always exist when $k=1$, but counterexamples exist when $k>1$. We will also describe the how the energy can be viewed as a regularization of the BV energy.

Mikhail Karpukhin, McGill University

**
Laplace eigenvalues and minimal surfaces in spheres
:**
We will give an overview of some recent estimates for Laplace eigenvalues on Riemannian surfaces. In particular, we will discuss the connection of optimal isoperimetric inequalities
with minimal surfaces and harmonic maps. Finally, this connection will be used in order to prove the sharp upper bound for all Laplace eigenvalues on the two-dimensional sphere. The talk is based on a joint work with N. Nadirashvili, A. Penskoi and I. Polterovich.

Peter Smillie, Harvard University

**
Entire spacelike surfaces of constant curvature in Minkowski 3-space
:**
We prove that every regular domain in Minkowski 3-space which is not a wedge contains a unique entire spacelike surface with constant intrinsic curvature equal to -1. This completes the classification of such surfaces in terms of their domains of dependence, for which partial results were obtained by Li, Guan-Jian-Schoen, and Bonsante-Seppi. Using this result, we obtain an analogous classification of entire spacelike surfaces with constant mean curvature (CMC). We'll apply these ideas to the Minkowski problem of prescribed curvature and to the construction CMC times in 2+1 relativity, and we'll see what we can say about the problem of deciding when the induced hyperbolic metric on an entire surface is complete. Everything is joint with Francesco Bonsante and Andrea Seppi.

Rafael Montezuma, Princeton University

**
A mountain pass theorem for minimal hypersurfaces with fixed boundary
:**
In this talk, we will be concerned with the existence of a third embedded minimal hypersurface, of mountain-pass type, spanning a closed submanifold B contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound B. In order to do so, we develop min-max methods similar to those of the recent work of De Lellis and Ramic, adapted to the discrete setting of Almgren and Pitts.

Pei-Ken Hung, Columbia University

**
The smoothing time of convex inverse mean curvature flows
:**
By using the local estimate recently proved by B. Choi and P. Daskalopoulos, we show that the smoothing time of convex inverse mean curvature flows is given by the smallest area of the initial tangent cone. As a corollary, convex inverse mean curvature flows on the sphere become smooth before the extinction time unless the corresponding cone splits a line. This is ongoing joint work with B. Choi.

Bing Wang, University of Wisconsin - Madison

**
The extension problem of the mean curvature flow in R^3
:**
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R^3. This is a joint work with H.Z. Li.

Xinliang An, U of Toronto

**
How to Make a Black Hole
:**
Black holes are predicted by Einstein's theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being. In this talk, with tools from hyperbolic PDE, quasilinear elliptic equations, geometric analysis and dynamical systems, we will prove that, through a nonlinear focusing effect, initially low-amplitude and diffused gravitational waves can give birth to a black hole region in our universe. This result extends the 1965 Penrose’s singularity theorem and it also proves a conjecture of Ashtekar on black-hole thermodynamics. Open problems and new directions will also be discussed.

Allen Brian, USMA West Point

**
Using IMCF to show stability of the Positive Mass Theorem and Riemannian Penrose Inequality
:**
In this talk we will discuss the stability of the Positive Mass Theorem (PMT) and Riemannian Penrose Inequality (RPI) for a sequence of compact regions of asymptotically flat manifolds, $U_T \subset M^3$, which are foliated by a uniformly controlled solution to Inverse Mean Curvature Flow (IMCF). The PMT result says that if the Hawking Mass of the outermost boundary of the regions, $U_T$, is approaching zero than the metric on $U_T$ is converging to the Euclidean metric in $L^2$. A similar result for the RPI will also be discussed where the metric on $U_T$ converges to the Schwarzschild metric in $L^2$. Key estimates will be introduced and ideas for how to extend this work will be explored.

Nicolau Aiex, University of British Columbia

**
The space of min-max hypersurfaces
:**
We use Lusternik-Schnirelmann Theory to study the topology of the space of closed embedded minimal hypersurfaces on a manifold of dimension between 3 and 7 and positive Ricci curvature. Combined with the works of Marques-Neves we can also obtain some information on the geometry of the minimal hypersurfaces they found.

Robin Neumayer, Northwestern University

**
The Cheeger constant of a Jordan domain without necks
:**
n 1970, Cheeger established lower bounds on the first eigenvalue of the Laplacian on compact Riemannian manifolds in terms of a certain isoperimetric problem. The analogous problem on domains of Euclidean space has generated much interest in recent years, due in part to its connections to capillarity theory, image processing, and landslide modeling. In this talk, based on joint work with Leonardi and Saracco, we give an explicit characterization of minimizers in this isoperimetric problem for a very general class of planar domains.

Kei Irie, Kyoto University

**
Denseness of closed geodesics on surfaces with generic Riemannian metrics
:**
We prove that, on a closed surface with a $C^\infty$-generic Riemannian metric, the union of nonconstant closed geodesics is dense. This result follows from a more general result about periodic orbits of Reeb dynamics on contact three-manifolds. The proof uses embedded contact homology (ECH), a version of Floer homology definedfor contact three-manifolds, which was introduced by Hutchings. In particular, the key ingredient is the ``Weyl law'' for ECH spectral numbers, which was proved by Cristofaro-Gardiner, Hutchings, and Ramos. We also discuss a denseness of minimal hypersurfaces for genericmetrics (joint work with Marques and Neves), which was obtained by applying a similar idea to the Weyl law for volume spectrum, which was proved by Liokumovich, Marques, and Neves.

Mathew Langford, The University of Tennessee

**
Ancient solutions of the mean curvature flow
:**
I will present a survey of existence and rigidity results for ancient solutions of mean curvature flow. In particular, I will describe recent work (with Theodora Bourni and Giuseppe Tinaglia) on the existence and uniqueness of rotationally symmetric ancient solutions which lie in a slab. Time permitting, we will finish by describing some interesting open problems.

Theodora Bourni, The University of Tennessee

**
Ancient Pancakes
:**
We show that, up to rigid motions, there is a unique compact, convex, rotationally symmetric, ancient solution of mean curvature flow that lies in a slab of width $\pi$ and in no smaller slab. This is joint work with Mat Langford and Giuseppe Tinaglia

Zahra Sinaei, Northwestern University

**
TBA
:**
In this talk, I discuss partial regularity of stationary solutions and minimizers u from a set \Omega\subset \R^n to a Riemannian manifold N, for the functional \int_\Omega F(x,u,|\nabla u|^2) dx. The integrand F is convex and satisfies some ellipticity, boundedness and integrability assumptions. Using the idea of quantitative stratification I show that the k-th strata of the singular set of such solutions are k-rectifiable.

Jesse Madnick, Stanford University

**
TBA
:**
TBA

Panagiotis Gianniotis, University of Toronto

**
The bounded diameter conjecture for 2-convex mean curvature flow.
:**
In this talk I address the bounded diameter conjecture for the mean curvature flow of smooth 2-convex hypersurfaces in $R^{n+1}$. In joint work with Robert Haslhofer, we prove that the intrinsic diameter of the evolving hypersurfaces is controlled, up to the first singular time, in terms of geometric information of the initial hypersurface. Moreover, this diameter estimate leads to sharp $L^{n−1}$ estimates for the curvature at each time.
Our estimates extend to mean curvature flow with surgery, which allows us to obtain the optimal $L^{n−1}$ estimate for any level set flow starting from a smooth 2-convex hypersurface. This improves the $L^{n−1−\varepsilon}$ curvature estimate that was previously established in work of Head and Cheeger-Haslhofer-Naber.

Antoine Song, Princeton University

**
Local min-max surfaces and existence of minimal Heegaard splittings
:**
Let M be a closed oriented 3-manifold not diffeomorphic to the 3-sphere, and suppose that there is a strongly irreducible Heegaard splitting H. Previously, Rubinstein announced that either there is a minimal surface of index at most one isotopic to H or there is a non-orientable minimal surface such that the double cover with a vertical handle attached is isotopic to H. He sketched a natural outline of a proof using min-max, however some steps are non-trivially incomplete and we will explain how to justify them. The key point is a version of min-max theory producing interior minimal surfaces when the ambient manifold has minimal boundary. Some corollaries of the theorem include the existence in any RP^3 of either a minimal torus or a minimal projective plane with stable universal cover. Several consequences for metric with positive scalar curvature are also derived.

Renato Bettiol, University of Pennsylvania

**
Non-uniqueness of conformal metrics with constant Q-curvature
:**
The problem of finding (complete) metrics with constant Q-curvature in
a prescribed conformal class is a famous fourth-order cousin of the
Yamabe problem. In this talk, I will provide some background on
Q-curvature and discuss how several non-uniqueness results for the
Yamabe problem can be transplanted to this context. However, special
emphasis will be given to multiplicity phenomena for constant
Q-curvature that have no analogues for the Yamabe problem, confirming
expectations raised by the lack of a maximum principle.

Otis Chodosh, Princeton University

**
Minimal surfaces in asymptotically flat 3-manifolds
:**
The study of minimal surfaces in asymptotically flat 3-manifolds goes back to the proof of the positive mass theorem by Schoen and Yau. I'll explain a rigidity theorem (joint with M. Eichmair) and an existence theorem (joint with D. Ketover) concerning such surfaces.

Chao Li, Stanford University

**
A polyhedron comparison theorem in 3-manifolds with positive scalar curvature
:**
We establish a comparison theorem for polyhedrons in 3-manifolds with nonnegative scalar curvature, answering affirmatively the dihedral rigidity conjecture by Gromov. For a large collections of polyhedrons with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less than those of the corresponding Euclidean model. We also establish the rigidity case.

Robert Haslhofer, University of Toronto

**
Minimal two-spheres in three-spheres
:**
We prove that any manifold diffeomorphic to S^3 and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in three-manifolds. Finally, we apply our methods to solve a problem posed by S.T. Yau in 1987, and to show that the assumptions in the multiplicity one conjecture and the equidistribution of widths conjecture are in a certain sense sharp. This is joint work with Dan Ketover.

Hannah Alpert, The Ohio State University

**
Morse broken trajectories and hyperbolic volume
:**
A large family of theorems all state that if a space is topologically complex, then the functions on that space must express that complexity, for instance by having many singularities. For the theorem in this talk, our preferred measure of topological complexity is the hyperbolic volume of a closed manifold admitting a hyperbolic metric (or more generally, the Gromov simplicial volume of any space). A Morse function on a manifold with large hyperbolic volume may still not have many critical points, but we show that there must be many flow lines connecting those few critical points. Specifically, given a closed n-dimensional manifold and a Morse-Smale function, the number of n-part broken trajectories is at least the Gromov simplicial volume. To prove this we adapt lemmas of Gromov that bound the simplicial volume of a stratified space in terms of the complexity of the stratification.

Jose Maria Espinar, IMPA - Brazil

**
Characterization of f-extremal disks
:**
In this talk we show uniqueness for overdetermined elliptic problems defined on topological disks with regular boundary, i.e., positive solutions $u$ to $\Delta u + f(u)=0$ in $\Omega \subset (M^2,g)$ so that $u = 0$ and $\frac{\partial u}{\partial \vec\eta} = cte $ along $\partial \Omega$, $\vec\eta$ the unit outward normal along $\partial\Omega$ under the assumption of the existence of a candidate family. In particular, this gives a positive answer to the Schiffer conjecture for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains, also called {\it Serrin Problem}, in $\mathbb S ^2$. This is a joint work with L. Mazet.

Gábor Székelyhidi, University of Notre Dame

**
New Calabi-Yau metrics on C^n
:**
I will discuss the construction of Calabi-Yau metrics on C^n with
maximal volume growth, with singular tangent cones at infinity. These
generalize recent examples constructed independently by Yang Li and
Conlon-Rochon

Fritz Hiesmayr, University of Cambridge

**
Index and spectrum of minimal hypersurfaces arising from
the Allen-Cahn construction
:**
The Allen-Cahn construction is a method for constructing
minimal surfaces of codimension 1 in closed manifolds.
In this approach, minimal hypersurfaces arise as the weak
limits of level sets of critical points of the Allen-Cahn
energy functional. This talk will relate the variational
properties of the Allen-Cahn energy to those of the area
functional on the surface arising in the limit, under the
assumption that the limit surface is two-sided.
In this case, bounds for the Morse indices of the critical
points lead to a bound for the Morse index of the limit
minimal surface. As a corollary, minimal hypersurfaces
arising from an Allen-Cahn p-parameter min-max construction
have index at most p. An analogous argument also establishes
a lower bound for the spectrum of the Jacobi operator of the
limit surface.

Shmuel Weinberger, The University of Chicago

**
Persistent Homology and Gromov's theorem on closed contractible geodesics.
:**
Gromov showed that on any closed Riemannian manifold whose fundamental group has unsolvable word problem, there are infinitely many closed nullhomotopic geodesics (of index 0). I will explain this theorem from the point of Persistent Homology of the free loop space, and then give some refinements (e.g. to less exotic fundamental groups) and extensions (to some other functionals other than energy on loops).

David Wiygul, UC Irvine

**
The Bartnik-Bray outer mass of small spheres
:**
In 1989 Robert Bartnik proposed a definition of quasilocal mass
in general relativity. The Bartnik mass is known to enjoy several
attractive properties but is not straightforward to evaluate. I will talk
about a first-order estimate for a natural modification of Bartnik's
definition applied to small perturbations of spheres in Euclidean space.
In particular I will describe an application to the small-sphere limit in
time-symmetric slices.

Nick Edelen , MIT

**
Quantitative Reifenberg for Measures
:**
In joint work with Aaron Naber and Daniele Valtorta, we
demonstrate a quantitative structure theorem for measures in R^n under
assumptions on the Jones \beta-numbers, which measure how close the
support is to being contained in a subspace. Measures with this property
have arisen in several interesting scenarios: in obtaining packing
estimates on and rectifiability of the singular set of minimal surfaces;
in characterizing L2-boundedness of Calderon-Zygmund operators; and as
an "annalist's" formulation of the travelling salesman problem.

Dan Ketover , Princeton University

**
Free boundary minimal surfaces of unbounded genus
:**
Free boundary minimal surfaces are natural variational objects
that have been studied since the 40s. In spite of this, very few explicit examples in
the simplest case of the round three ball are known. I will describe how variational
methods can be used to construct new examples with unbounded genus resembling a
desingularization of the critical catenoid and flat disk. I will also give a new variational
interpretation of the previously known examples.

Marco Radeschi , University of Notre Dame

**
Minimal hypersurfaces in compact symmetric spaces
:**
A conjecture of Marques-Neves-Schoen says that for every
embedded minimal hypersurface M in a manifold of positive Ricci curvature,
the first Betti number of M is bounded above linearly by the index of M.
We will show that for every compact symmetric space this result holds, up
to replacing the index of M with its extended index. Moreover, for special
symmetric spaces, the actual conjecture holds for all metrics in a
neighbourhood of the canonical one. These results are a joint work with R.
Mendes.

Pierre Albin , University of Illinois at Urbana-Champaign

**
Analytic torsion of manifolds with fibered cusps
:**
Analytic torsion is a spectral invariant of the Hodge Laplacian
of a manifold with a flat connection. On a closed manifold it is equal to
a topological invariant known as Reidemeister torsion. I will describe joint
work with Frédéric Rochon and David Sher establishing a topological
expression for the analytic torsion of a manifold with fibered cusp ends
(such as a locally symmetric space of rank one). We establish our result by
controlling the behavior of the spectrum along a degenerating class of
Riemannian metrics.

Jacob Bernstein , Johns Hopkins University

**
Surfaces of Low Entropy
:**
Following Colding and Minicozzi, we consider the entropy of
(hyper)-surfaces in Euclidean space. This is a numerical measure of the
geometric complexity of the surface and is intimately tied to to the
singularity formation of the mean curvature flow. In this talk, I will
discuss several results that show that closed surfaces for which the
entropy is small are simple in various senses. This is all joint work with
L. Wang.

Pedro Gaspar , IMPA

**
Minimal hypersurfaces and the Allen-Cahn equation on closed manifolds
:**
Since the late 70s parallels between the theory of phase transitions and critical points of the area functional have helped us to understand variational properties of certain semi-linear elliptic PDEs and spaces of hypersurfaces which minimize the area in an appropriate sense. We will discuss some recent developments in this direction which extend well-known analogies regarding minimizers to more general variational solutions. In particular, borrowing ideas from the min-max theory of minimal hypersurfaces, we study the number of solutions of the Allen-Cahn equation in a closed manifolds and solutions with least non-trivial energy.

Or Hershkovits , Stanford University

**
Uniqueness of mean curvature flow with mean convex singularities
:**
Given a smooth compact hypersurface in Euclidean space, one can
show that there exists a unique smooth evolution starting from it,
existing for some maximal time. But what happens after the flow becomes
singular? There are several notions through which one can describe weak
evolutions past singularities, with various relationship between them.
One such notion is that of the level set flow. While the level set flow is
almost by definition unique, it has an undesirable phenomenon called
fattening: Our "weak evolution" of n-dimensional hypersurfaces may develop
(and does develop in some cases) an interior in R^{n+1}. This fattening
is, in many ways, the right notion of non-uniqueness for weak mean
curvature.
As was alluded to above, fattening can not occur as long as the flow is
smooth. Thus it is reasonable to say that the source of fattening is
singularities. Permitting singularities, it is very easy to show that
fattening does not occur if the initial hypersurface, and thus all the
evolved hypersurface, are mean convex. Thus, singularities encountered
during mean convex mean curvature flow should be of the kind that does not
create singularities (i.e, the local structure of the singularities should
prevent fattening, without any global mean convexity assumption). To put
differently, it is reasonable to conjecture that:
"An evolving surface cannot fatten unless it has a singularity with no
spacetime neighborhood in which the surface is mean convex".
In this talk, we will phrase a concrete formulation of this conjecture,
and describe its proof. This is a joint work with Brian White.

Xin Zhou , MIT

**
Min-max minimal hypersurfaces with free boundary
:**
I will present a joint work with Martin Li. Minimal surfaces
with free boundary are natural critical points of the area functional in
compact smooth manifolds with boundary. In this talk, I will describe a
general existence theory for minimal surfaces with free boundary. In
particular, I will show the existence of a smooth embedded minimal
hypersurface with free boundary in any compact smooth Euclidean domain. The
minimal surfaces with free boundary were constructed using the min-max
method.
Our result allows the min-max free boundary minimal hypersurface to be
improper; nonetheless the hypersurface is still regular.

Henrik Matthiesen , Max Plank Institute for Mathematics Bonn

**
Existence of metrics maximizing the first eigenvalue on closed
surfaces
:**
We show that on each closed surface of fixed topological type,
orientable or non-orientable, there is a metric, smooth away from finitely many conical singularities,
that maximizes the first eigenvalue among all unit area metrics. The key new ingredient are
several monotonicity results relating the corresponding maximal eigenvalues. This is joint work with Anna Siffert.

Gang Liu, Northwestern University

**
On some recent progress of Yau's uniformization conjecture
:**
Yau's uniformization conjecture states that a complete noncompact Kahler manifold with positive bisectional
curvature is biholomorphic to the complex Euclidean space. We shall discuss some recent progress via the Gromov-Hausdorff
convergence technique.

John Ma, University of British Columbia

**
**** Compactness, finiteness properties of Lagrangian self-shrinkers in R^4 and Piecewise mean curvature flow.
:**
In this talk, we discuss a compactness result on the space of compact Lagrangian self-shrinkers in R^4. When the area is bounded above
uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a
Lojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori. Using the finiteness of entropy values, we construct a
piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in R^4, along which the Lagrangian condition is preserved, area is
decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian
version of the construction for embedded surfaces in R^3 by Colding and Minicozzi. This is a joint work with Jingyi Chen.

Laura Schaposnik, University of Illinois at Chicago

**
On the geometry of branes in the moduli space of Higgs bundles.
:**
After giving a gentle introduction to Higgs bundles, their moduli space and the associated Hitchin fibration,
I shall describe how actions both on groups and on surfaces (anti-holomorphic or of finite groups) lead to families of interesting
subspaces of the moduli space of Higgs bundles (the so-called branes). Finally, we shall look at correspondences between these branes
that arise from Langlands duality, as well as from other relations between Lie groups.

Pei-Ken Hung, Columbia University

**
**** A Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold
:**
We prove a sharp inequality for hypersurfaces in the
Anti-deSitter-Schwarzschild manifold. This inequality generalizes the
classical Minkowski inequality for surfaces in the three dimensional
Euclidean space, and has a natural interpretation in terms of the Penrose
inequality for collapsing null shells of dust. The proof relies on a
monotonicity formula for inverse mean curvature flow, and uses a geometric
inequality established by Brendle.

Lan-Hsuan Huang, University of Connecticut

**
Geometry of Static Asymptotically Flat 3-Manifolds
:**
A static potential on a 3-manifold reflects the symmetry of the spacetime development and is
also closely related to the scalar curvature deformation.
We discuss how minimal surfaces of 3-manifold interact with the static potential and, as an application,
give a new proof to the rigidity of the Riemannian positive mass theorem.

Yu Li, University of Wisconsin-Madison

**
Ricci flow on asymptotically Euclidean manifolds
:**
In this talk, we prove that if an asymptotically Euclidean (AE) manifold with nonnegative scalar curvature has long
time existence of Ricci flow, it converges to the Euclidean space in the strong sense. By convergence, the mass will drop to
zero as time tends to infinity. Moreover, in three dimensional case, we use Ricci flow with surgery to give an independent
proof of positive mass theorem.

Yannick Sire, Johns Hopkins University

**
Bounds on eigenvalues on Riemannian surfaces
:**
In a joint program with N. Nadirashvili, we developed some tools to prove
existence of extremal metrics in conformal classes for eigenvalues of the Laplace-Beltrami operator on surfaces.
I will describe these results and move to an application to the isoperimetric inequality of the third eigenvalue on the 2-sphere.

Henri Roesch, Duke University

**
Proof of a Null Penrose conjecture using a new Quasi-local Mass
:**
We define an explicit quasi-local mass functional which is non-decreasing along all
foliations (satisfying a convexity assumption) of null cones. We use this new functional to prove the null Penrose conjecture
under fairly generic conditions.

Luca Spolaor, MIT

**
A direct approach to an epiperimetric inequality for Free-Boundary problems
:**
Using a direct approach, we prove a 2-dimensional epiperimetric inequality for the one-phase problem in the scalar and vectorial cases and
for the double-phase problem. From this we deduce the $C^{1,\alpha}$ regularity of the free-boundary in the scalar one-phase and double-phase
problems, and of the reduced free-boundary in the vectorial case, without any restriction on the sign of the component functions. In this talk I
will try to explain the proof of the epiperimetric inequality in the scalar one-phase problem. This is joint work with Bozhidar Velichkov.

Victoria Sadovskaya, Department of Mathematics - Penn State

**Boundedness, compactness, and invariant norms for operator-valued cocycles.:**
We consider group-valued cocycles over dynamical systems with
hyperbolic behavior, such as hyperbolic diffeomorphisms or subshifts
of finite type. The cocycle A takes values in the group of invertible
bounded linear operators on a Banach space and is Holder continuous.
We consider the periodic data of A, i.e. the set of its return values
along the periodic orbits in the base. We show that if the periodic data
of A is bounded or contained in a compact set, then so is the cocycle.
Moreover, in the latter case the cocycle is isometric with respect to
a Holder continuous family of norms. This is joint work with B. Kalinin.

Baris Coskunuser , Boston College

**
Embeddedness of the solutions to the H-Plateau Problem
:**
In this talk, we will give a generalization of Meeks and Yau's embeddedness result on the solutions of the Plateau
problem to the constant mean curvature disks. arXiv:1504.00661

Lu Wang, Department of Mathematics - University of Wisconsin-Madison

**Asymptotic structure of self-shrinkers:**
We show that each end of a noncompact self-shrinker in Euclidean
3-space of finite topology is smoothly asymptotic to a regular cone or a
self-shrinking round cylinder.

Spencer Becker-Kahn, Department of Mathematics - MIT

**Zero Sets of Smooth Functions and p-Sweepouts:**
It is well known that any closed set can occur as the zero set of a smooth function. But in many contexts in analysis, one does not work with arbitrary smooth functions; one works with smooth functions that vanish to only finite order. And the zero set of any such function must have some regularity and structure.
With recourse to an analogy made by Gromov and some recent conjectures of Marques and Neves, I will explain the application of some general results about such zero sets to p-sweepouts (which are p-dimensional families of generalized hypersurfaces that sweepout a smooth manifold in a certain sense). In the course of doing so, I will explain a new result about the regularity of zero sets of smooth functions near a point of finite vanishing order (joint work with Tom Beck and Boris Hanin)..

Jonathan Zhu, Department of Mathematics - Harvard University

**
Entropy and self-shrinkers of the mean curvature flow:**
The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropy-stable self-shrinkers that may have a small singular set.

Greg Chambers, Department of Mathematics - The University of Chicago

**
Existence of minimal hypersurfaces in non-compact manifolds:**
I prove that every complete non-compact manifold of finite volume admits a minimal hypersurface of finite area. This work
is in collaboration with Yevgeny Liokumovich.

Peter McGrath, Brown University

**
New doubling constructions for minimal surfaces:**
I will discuss recent work (with Nikolaos Kapouleas) on constructions of new embedded minimal surfaces using singular perturbation
'gluing' methods. I will discuss at length doublings of the equatorial S^2 in S^3. In contrast to earlier work of Kapouleas, the catenoidal bridges
may be placed on arbitrarily many parallel circles on the base S^2. This necessitates a more detailed understanding of the linearized problem and
more exacting estimates on associated linearized doubling solutions.

Franco Vargas Pallete, UC Berkeley

**
Renormalized volume:**
The renormalized volume V_R is a finite quantity associated to certain hyperbolic 3-manifolds of infinite volume. In this talk I'll discuss its definition and some properties for acylindrical manifolds, namely local convexity and convergence under geometric limits. If time suffices I'll also discuss some applications and further examples.

Christos Mantoulidis, Stanford University

**
Fill-ins, extensions, scalar curvature, and quasilocal mass:**
There is a special relationship between the Jacobi and the ambient scalar curvature operator. I'll talk about an extremal bending technique that exploits this relationship. It lets us compute the Bartnik mass of apparent horizons and disprove a form of the Hoop conjecture due to G. Gibbons. Then, I'll talk about a derived "cut-and-fill" technique that simplifies 3-manifolds of nonnegative scalar curvature. It is used in studying a priori L^1 estimates for boundary mean curvature for compact initial data sets, and in generalizing Brown-York mass. Parts of this talk reflect work done jointly with R. Schoen/P. Miao.

Hung Tran, UC Irvine

**
Index Characterization for Free Boundary Minimal Surfaces:**
A FBMS in the unit Euclidean ball is a critical point of the area functional among all surfaces with boundaries in the unit sphere, the boundary of the ball. The Morse index gives the number of distinct admissible deformations which decrease the area to second order. In this talk, we explain how to compute the index from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. We also discuss applications to a conjecture about FBMS with index 4.

For further information on this seminar or this webpage, please contact Marco Guaraco at
`guaraco(at)math.uchicago.edu`, Lucas Ambrozio at
`lambrozio(at)math.uchicago.edu` or
Rafael Montezuma at `montezuma(at)math.uchicago.edu`